A new optimal digital halftoning technique based on the discrete cosine transform
Introduction
Many displaying devices are basically bilevel in nature. The displaying cell is either on or off, bright or dark, white or black. Digital halftoning is a technique that displays a gray-level image with two levels [1], [6], [16]. Digital halftoning techniques can be done in the spatial domain or in the frequency domain. The techniques can be classified into the point methods [9], [10], [11], [12], [13], the neighborhood methods [2], [4], [5], [7] and the search methods [3], [8], [17]. The point methods and the neighborhood methods are done in the spatial domain, while the search methods can be done either in the spatial domain [14], [15] or in the frequency domain [3], [8], [17].
In the point methods, Bayer [2], Lippel [9], [11], [12] and Limb [10] developed their methods for two level rendition of continuous image. Mitsa and Parker [13] used a blue-noise mask to halftone a gray-level image. In the neighborhood methods, error diffusion [5] is a well-known method to solve the false contouring problem. It uses neighborhood operations to diffuse the error over a weighted neighborhood. The shortcoming of this algorithm is that the error between each bilevel pixel and its original pixel is dispersed to its neighborhood. Eschbach and Knox [4] proposed an error diffusion algorithm with edge enhancement.
In the search methods, Pappas [14], [15] used a least-squares model-based halftoning method for both B&W and color printers which incorporates the properties of the display device and human visual system in the spatial domain. Carnevali et al. [3] used simulated annealing and Kollias [8] used a unified network for digital image halftoning based on the minimization of the weighted mean squared error in the frequency domain of the discrete Fourier transform. Zakhor [17] also proposed a new class of digital halftoning techniques with linear programming based on the discrete Fourier transform. Instead of using the discrete Fourier transform, most video coding techniques such as JPEG, MPEG I, MPEG II and HDTV use the discrete cosine transform (DCT). Since the DCT just computes real numbers it is much faster than the discrete Fourier transform, which computes complex numbers. This paper proposes a new digital halftoning algorithm based on the DCT. By our algorithm, the image can be divided into small subimages in order to reduce computational complexity. An optimal bilevel subimage is obtained by minimizing the distortion between the bilevel subimage and its corresponding gray-level subimage in the DCT domain. The optimization exhaustively searches for the optimal bilevel subimage with the least mean square error to replace the gray-level subimage. By our experimental results, not only are the edges in the halftoned image enhanced, but also the false contours are greatly reduced.
Section snippets
The proposed algorithm
Given a gray-level image, a digital halftoning algorithm can be stated as below.
Find a bilevel image that gives the illusion of a gray-level image on a bilevel display.
In the bilevel image, we shall call the pixel “1-pixel” if its gray-level is equal to 1; otherwise it is called “0-pixel”. There are two problems to find an optimal bilevel image. One is how many 1-pixels are needed in the optimal bilevel image. The other is how to distribute these 1-pixels over the image. For the first question,
Simulation
Four images of 512×512 are halftoned and printed on a laser printer with . Fig. 2 shows the halftoned images based on the proposed algorithm with the weighting coefficients given in Eq. (10). These bilevel images are very smooth without contouring effect.
Conclusions
We proposed a new binary optimization technique for digital halftoning. The proposed algorithm minimizes the distortion between the gray-level image and the halftoned image in the DCT domain. It is much faster than Zakhor's algorithm. It greatly reduces false contouring, and produces a very smooth halftoned image.
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